91Ó°ÊÓ

A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in seconds) can be modeled by $$h(t)=53+50 \sin \left(\frac{\pi}{10} t-\frac{\pi}{2}\right)$$ (a) Find the period of the model. What does the period tell you about the ride? (b) Find the amplitude of the model. What does the amplitude tell you about the ride? (c) Use a graphing utility to graph one cycle of the model.

The daily consumption \(C\) (in gallons) of diesel fuel on a farm is modeled by $$C=30.3+21.6 \sin \left(\frac{2 \pi t}{365}+10.9\right)$$ where \(t\) is the time (in days), with \(t=1\) corresponding to January 1. (a) What is the period of the model? Is it what you expected? Explain. (b) What is the average daily fuel consumption? Which term of the model did you use? Explain. (c) Use a graphing utility to graph the model. Use the graph to approximate the time of the year when consumption exceeds 40 gallons per day.

An airplane, flying at an altitude of 6 miles, is on a flight path that passes directly over an observer (see figure). Let \(\theta\) be the angle of elevation from the observer to the plane. Find the distance \(d\) from the observer to the plane when (a) \(\theta=30^{\circ},\) (b) \(\theta=90^{\circ}\) and \((c) \theta=120^{\circ}.\)

Fill in the blank. If not possible, state the reason. As \(x \rightarrow-\infty,\) the value of arctan \(x \rightarrow\) \(\square\).

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi),\) and sketch the graph of the inverse trigonometric function.

Define the inverse secant function by restricting the domain of the secant function to the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi],\) and sketch the graph of the inverse trigonometric function.